String Girdling Earth Calculator
Solve string girdling earth problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Formula
Extra String = 2 * pi * h
The extra string needed to raise a string uniformly by height h above any sphere equals exactly 2 * pi * h. Remarkably, this formula is independent of the sphere radius, meaning the same amount of extra string works for any size sphere.
Worked Examples
Example 1: Classic String Around Earth Problem
Problem: A string is wrapped around the Earth (radius 6,371 km). How much extra string is needed to raise it 1 meter uniformly above the surface?
Solution: Original circumference = 2 * pi * 6,371,000 = 40,030,174 m\nNew circumference = 2 * pi * 6,371,001 = 40,030,180.28 m\nExtra string = 2 * pi * 1 = 6.2832 m\nNotice: the radius cancels completely!\nThe same 6.28 m works for ANY sphere.
Result: Extra string = 6.2832 meters (only 2*pi*h, independent of Earth radius)
Example 2: Person Walking Under the String
Problem: How much extra string is needed to raise the string 2 meters above Earth so a person could walk under it?
Solution: Extra string = 2 * pi * h = 2 * pi * 2 = 12.566 m\nOriginal circumference = 40,030,174 m\nNew circumference = 40,030,186.57 m\nPercentage increase = 12.566 / 40,030,174 = 0.0000314%\nJust 12.6 meters of extra string!
Result: Extra string = 12.566 meters for a 2-meter gap all around Earth
Frequently Asked Questions
What is the String Girdling Earth problem?
The String Girdling Earth problem is a famous mathematical puzzle that reveals a counterintuitive result about circles and circumferences. Imagine a string wrapped tightly around the Earth at the equator. If you wanted to raise the string uniformly by 1 meter above the surface all the way around, how much extra string would you need? Most people guess thousands of kilometers, but the answer is only about 6.28 meters (2 * pi meters). This tiny amount of extra string is independent of the original circle size, meaning the same 6.28 meters would lift the string 1 meter above a basketball, a planet, or even the Sun. The problem has been discussed in mathematics since at least the 1700s.
Why is the extra string independent of the sphere radius?
The independence from radius is the key mathematical insight and can be shown algebraically. The original circumference is C1 = 2 * pi * r, and the new circumference at height h above the surface is C2 = 2 * pi * (r + h). The extra string needed is C2 - C1 = 2 * pi * (r + h) - 2 * pi * r = 2 * pi * r + 2 * pi * h - 2 * pi * r = 2 * pi * h. The radius r cancels completely, leaving only the term 2 * pi * h, which depends solely on the desired gap height. This means lifting a string 1 meter above Earth requires the same extra 6.283 meters as lifting it 1 meter above a marble. This algebraic cancellation is what makes the result so surprising and unintuitive.
How much extra string do you need for different gap heights?
Since the extra string formula is simply 2 * pi * h, the calculation is straightforward for any gap height. For a 1-centimeter gap: 2 * pi * 0.01 = 0.0628 meters (about 6.3 cm). For a 10-centimeter gap: 0.628 meters. For a 1-meter gap: 6.283 meters. For a 2-meter gap (person walking under): 12.566 meters. For a 10-meter gap: 62.83 meters. For a 100-meter gap: 628.3 meters. Notice the perfectly linear relationship. Every additional meter of gap height requires exactly 2 * pi (approximately 6.283) meters of extra string, regardless of whether the original sphere is a marble or Jupiter.
How do plate tectonics shape the Earth's surface?
Earth's lithosphere is divided into tectonic plates that move on the asthenosphere. Divergent boundaries create new crust (mid-ocean ridges), convergent boundaries destroy crust (subduction zones) or build mountains, and transform boundaries cause earthquakes. Plates move 1-10 cm per year, driven by mantle convection.
Is String Girdling Earth Calculator free to use?
Yes, completely free with no sign-up required. All calculators on NovaCalculator are free to use without registration, subscription, or payment.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.